Timely Persuasion with Opposing Senders: When Does Competition Help the Receiver?

Timely Persuasion with Opposing Senders: When Does Competition Help the Receiver?

arXiv:2610.04468v1 [cs.IT] 3 Oct 2026

Ahmet Bugra Gundogan and Melih Bastopcu Abstract— Status updates let a receiver track a changing process, but a sender that controls when they are sent can steer the receiver’s estimate without misreporting. We study competition between two such senders with opposing preferences, who report on a binary continuous-time Markov chain using state-dependent Poisson rates under individual budgets. The receiver follows the reports through a zero-order-hold estimator only if this yields at least the long-run weighted correct-estimation utility of a prior-based default estimate. We model the interaction as a Stackelberg game in which the sender favored by this default leads, the opposing sender follows, and the receiver then selects its estimator. We characterize the purestrategy equilibria in closed form. When the follower can obtain positive utility against a silent leader, limited state-0 reporting by the leader merely frees part of the follower’s budget for state-1 reports, so every optimal leader policy is either silence or full-budget state-0 reporting, depending on explicit budget thresholds. A single sender with the same total budget leaves the receiver at its default utility, whereas the full-budget intervention outcome makes the receiver’s participation constraint slack and strictly improves its utility. At the intervention threshold, silence and full-budget intervention yield identical sender payoffs but different receiver utilities. Numerical results illustrate these regimes and compare equilibrium performance with a receiver-optimal reporting benchmark.

I. I NTRODUCTION When a receiver tracks a changing process through status updates, its estimate is only as current as the latest update. This has made the age of information a central performance measure [1], [2], and keeping information fresh is the natural goal when the sender wants the receiver to be accurate. A sender that instead has a stake in the receiver’s estimate can use timing to persuade without misreporting. In other words, by reporting favorable states promptly and unfavorable ones less frequently, it lets the receiver’s information go stale precisely when the state turns unfavorable. Since the receiver can ignore the updates altogether, the sender must still report unfavorable states often enough to keep the receiver following them [3]. A second sender with the opposite preference can supply these reports itself, but doing so also frees the first sender to report favorable states even more often. This raises two questions: when does the opposing sender choose to intervene, and when does its intervention improve the receiver’s estimation performance? We address these questions through a game in which two senders continuously observe a binary continuous-time A.B. Gundogan and M. Bastopcu are with the Department of Electrical and Electronics Engineering, Bilkent University, Ankara, Türkiye.

{bugra.gundogan, bastopcu}@bilkent.edu.tr This work was supported in part by the TUBITAK 2232-B program (Project No:124C533).

leader sender 2

0

µ λ

(s1,c1) 1

x(t)

binary CTMC source

receiver’s estimate (s2,c2)

x̂(t) = 1

receiver m(t)

or x̂(t) = 0

sender 1 follower

Fig. 1: System model with two competing senders and one receiver. Markov chain (CTMC) and send truthful reports of its state to a common receiver as shown in Fig. 1. Each sender’s strategy is a pair of state-dependent Poisson reporting rates, constrained by an individual budget on their sum. Irrespective of the source state, sender 1 is rewarded whenever the receiver’s estimate is 1 and sender 2 whenever it is 0, so their payoffs are constant-sum. The receiver maximizes its long-run weighted correct-estimation utility by choosing between a zero-order-hold (ZOH) estimator, which retains the most recently reported state, and a default constant estimate chosen optimally under the stationary prior; throughout, we consider the regime in which this default estimate is 0. The receiver therefore follows the reports only if ZOH estimation yields at least its default utility, a requirement we call the participation constraint (PC). Since this default already favors sender 2, we let it act as the leader and ask when it chooses to intervene. The resulting Stackelberg game has three stages: sender 2 commits to its reporting policy; sender 1, the follower, observes it and commits to its own; and the receiver observes both and selects its estimator. Both senders then report concurrently under these fixed policies. The receiver does not use information conveyed by the absence of reports, and our equilibrium and receiver-benefit results apply within this estimator architecture. Within the age-of-information literature, our model is closest to remote estimation, where sampling policies are designed to improve the receiver’s estimate of Wiener processes [4], binary infection status [5], and finite-state CTMCs under sampling rate constraints [6]. In our problem, the reporting rates are chosen by senders with opposing preferences over this estimate, so estimation performance is an equilibrium outcome rather than a design objective. As in Bayesian persuasion [7] and information design [8], our senders commit to their policies in advance. Dynamic persuasion has been studied with partially informed receivers [9], Markovian states [10], [11], and stochastic state revelations [12]. Ely [13] studies dynamic persuasion through messages about an evolving stochastic process, with a re-

ceiver that acts on its beliefs. Related formulations address sequential disclosure for change detection [14], continuoustime persuasion through filtering [15], and costly information provision without commitment to future actions [16]. In contrast to these models, which allow the sender to shape the receiver’s information through signal or disclosure design, our senders cannot alter the content of information and control only the timing of truthful state reports. Our prior work [3] studies this form of timely persuasion with a single sender, and we retain its source, reporting, and receiver models. There, whenever persuasion is feasible, the sender reports state 0 just often enough to make the PC bind, so the receiver gains nothing from following the reports. Asking whether competition can improve on this outcome connects our work to multi-sender persuasion, where the effect of competition on information revelation depends on the available signal structures [17], [18]. Au and Kawai [19] show that increasing the number of symmetric senders leads to more aggressive disclosure in their model, and Li and Norman [20] characterize equilibrium outcomes under sequential persuasion. In the control literature, Velicheti et al. [21] study simultaneous sender commitments with Gaussian information and quadratic objectives, including a dynamic extension, and establish a receiver benefit from competition relative to cooperation among the senders. Our senders instead commit sequentially to the rates of truthful reports. Since J1 + J2 = 1, maximizing the sum of sender payoffs does not select a reporting policy. We therefore compare the receiver’s equilibrium utility with that under a single sender with the same total budget. We answer both questions posed at the outset. For the first, we derive the follower’s best response and characterize the pure-strategy Stackelberg equilibria in closed form. When the follower can obtain positive utility on its own, the leader’s state-0 reports first help the follower by freeing its budget for state-1 reports and hurt it only once they alone exceed what participation requires. The leader therefore chooses between silence and full-budget state-0 reporting. We identify a threshold on the follower’s budget below which silence is optimal regardless of the leader’s budget, and a second threshold governing intervention above it. For the second, whenever the full-budget intervention outcome is optimal, the leader’s state-0 reports make the PC slack and strictly increase the receiver’s utility. In contrast, the optimal singlesender benchmark leaves the receiver at its default utility. At the intervention threshold, the two optimal leader policies yield identical sender payoffs but different receiver utilities. Numerical results illustrate these regimes and show that an appropriate budget division can attain the receiver-optimal reporting benchmark for the parameters considered. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION We consider a binary source observed by two senders that report to a receiver, as shown in Fig. 1. The source state x(t) ∈ {0, 1} evolves as a continuous-time Markov chain (CTMC), with transition rate λ > 0 from state 0 to state 1 and rate µ > 0 from state 1 to state 0. Thus, the source has

a unique stationary distribution, given by µ λ π0 = , π1 = . (1) λ+µ λ+µ Both senders continuously observe the source. The receiver does not observe it directly but knows the transition rates λ and µ. The receiver wants to track the source’s state as accurately as possible. To do so, it forms an estimate x̂(t) ∈ {0, 1} at time t and obtains an instantaneous utility as uR (x, x̂) = q1{x=0,x̂=0} + (1 − q)1{x=1,x̂=1} , (2) where q ∈ (0, 1) weights correct estimates of the two states and 1{·} is the indicator function. Constant estimates 0 and 1 yield average utilities qπ0 and (1 − q)π1 , respectively. Throughout the paper, we assume qπ0 > (1 − q)π1 , so the receiver’s optimal constant estimate is 0. Sender 1 prefers the receiver’s estimate to be 1, whereas sender 2 prefers it to be 0, regardless of the source state: u1 (x, x̂) = 1{x̂=1} ,

u2 (x, x̂) = 1{x̂=0} .

(3)

Let sj and cj denote the reporting rates of sender j ∈ {1, 2} when x(t) = 1 and x(t) = 0, respectively. Sender j selects the stationary reporting policy βj = (sj , cj ). Conditional on the source trajectory, each sender’s reports form a Poisson process with these state-dependent rates, and the two reporting processes are independent. The rates depend only on the current source state, not on past reports or the receiver’s estimate. Each report contains the current source state and arrives without delay or error. Following [3], we impose the conditional-intensity budget βj ∈ Bj := {(sj , cj ) ∈ R2≥0 : sj + cj ≤ Rj },

(4)

where Rj ≥ 0 is sender j’s reporting capacity. With the constraints on the sum of the conditional intensities, the longrun average reporting rate of sender j is π1 sj + π0 cj . The source parameters, budgets, and utility functions are common knowledge. Following [3, Remark 1], the receiver chooses between two estimation policies. Under σs , it uses the reports from both senders where each new report replaces the current estimate with the reported source state, which is then held until the next report. Under σd , it ignores all reports and always estimates 0. Let tk denote the kth report arrival from either sender. Then, under σs , the receiver’s estimate x̂(t) becomes x̂(t) = x(tk ),

tk ≤ t < tk+1 .

(5)

Before the first report, the receiver estimates 0. We study a receiver with a fixed hold-last-value interface that can either be enabled or disabled before operation. Accordingly, the receiver selects one of the two admissible policies, σs or σd , before reporting begins and uses it thereafter. The participation constraint introduced below compares these two modes which does not imply that the held estimate is optimal after every observation history. We model the interaction as a three-stage Stackelberg game with sender 2 as the leader and sender 1 as the

follower. In the first stage, sender 2 commits to a reporting policy β2 ∈ B2 . In the second stage, sender 1 observes β2 and commits to a reporting policy β1 ∈ B1 . In the third stage, the receiver observes (β1 , β2 ) and selects an estimator σ ∈ {σs , σd }. These choices remain fixed throughout the subsequent reporting and estimation. Because the preferred estimate of sender 2 coincides with the receiver’s default estimate 0, the leader effectively decides whether to intervene against sender 1, which prefers estimate 1, anticipating the subsequent choices of sender 1 and the receiver. The payoff of player k ∈ {1, 2, R} is its long-run average utility "Z # T 1 E uk (x(t), x̂(t)) dt . (6) Jk (β1 , β2 , σ) = lim T →∞ T 0 Since u1 (x, x̂)+u2 (x, x̂) = 1 for all (x, x̂), the senders’ payoffs satisfy J1 + J2 = 1 for every strategy profile (β1 , β2 , σ). The game between the senders is therefore constant-sum and strategically equivalent to a zero-sum game in which sender 2 minimizes the payoff that sender 1 attains through its best response. A. Average Utilities and the Participation Constraint Under σs , the estimate is updated at the report arrivals of both senders. Since the two reporting processes are conditionally independent given the source trajectory, their superposition is conditionally Poisson with rate s = s1 + s2 in state 1 and c = c1 + c2 in state 0. Because every report reveals the current source state, the ZOH estimate depends only on the arrival times of this aggregate process and not on which sender generated each report. Hence, (x(t), x̂(t)) is a CTMC on {0, 1}2 with the same transition structure as the single-sender model in [3, Sec. II-A], with (s, c) in place of that sender’s reporting rates. Consequently, the payoffs under σs depend on (β1 , β2 ) only through (s, c), and, with a slight abuse of notation, we write Jk (s, c) := Jk (β1 , β2 , σs ) for k ∈ {1, 2, R}. For (s, c) 6= (0, 0), this chain has a unique stationary distribution, and evaluating the payoffs under it as in [3, Sec. II-A] gives JR (s, c) =

qµc(µ + s) + (1 − q)λs(λ + c) , (λ + µ)(µc + λs + cs)

(7)

λs(c + λ + µ) , (λ + µ)(µc + λs + cs)

(8)

J1 (s, c) =

and J2 (s, c) = 1 − J1 (s, c). When s = c = 0, no reports arrive and the estimate remains at its initial value 0, so JR (0, 0) = qπ0 , J1 (0, 0) = 0, and J2 (0, 0) = 1. Under σd , the receiver’s estimate is identically 0 for all t, so JR = qπ0 , J1 = 0, and J2 = 1 for every (β1 , β2 ). Since the receiver obtains qπ0 under σd and JR (s, c) under σs , following the reports is optimal within the receiver’s admissible policy set if and only if the participation constraint (PC) JR (s, c) ≥ qπ0 (9) holds. We define the term cmin := qµ/(1 − q) − λ, which is positive because qπ0 > (1 − q)π1 . Subtracting qπ0 from (7)

and using (1 − q)(λ + c) − qµ = (1 − q)(c − cmin ) gives, for (s, c) 6= (0, 0), JR (s, c) − qπ0 =

(1 − q)λs(c − cmin ) . (λ + µ)(µc + λs + cs)

(10)

Hence, for s > 0, the PC is equivalent to c1 + c2 ≥ cmin ,

(11)

in agreement with [3, eq. (10)], whereas for s = 0 it holds with equality. When s > 0, participation therefore requires a minimum aggregate intensity cmin of state-0 reports. In the single-sender model, this requirement must be met by that sender alone. Here, it is met jointly by the state-0 reports of both senders. When the PC holds with equality, the receiver is indifferent between σs and σd . We adopt the tie-breaking rule under which the receiver selects σs at indifference unless s = c = 0, in which case it selects σd . The receiver’s response is thus ( σs , JR (s, c) ≥ qπ0 and (s, c) 6= (0, 0), σ ∗ (β1 , β2 ) = (12) σd , otherwise, where (s, c) are the aggregate rates of the senders defined previously. This rule favors sender 1 and ensures that its maximum is attained when the PC binds. Under the opposite rule, a follower facing a silent leader with R1 > cmin would obtain a positive payoff only if c1 > cmin , so its payoff would approach its supremum as c1 approaches cmin from above, without attaining it. B. Stackelberg Game Formulation Since sender 1 observes β2 and the receiver observes (β1 , β2 ), their strategies are maps β1∗ : B2 → B1 and σ ∗ : B1 × B2 → Σ, where Σ := {σs , σd }. Definition 1: The tuple (β1∗ , β2∗ , σ ∗ ) is a pure-strategy Stackelberg equilibrium if, for all β1∈ B1 , β2∈ B2 , and σ∈ Σ,
(13) JR β1 ,β2 , σ ∗ (β1 , β2 ) ≥ JR (β1 , β2 , σ),

∗ ∗ ∗ ∗ J1 β1 (β2 ),β2 ,σ (β1 (β2 ), β2 ) ≥ J1 β1 , β2 , σ (β1 , β2 ) , (14)
J2 β1∗ (β2∗ ), β2∗ , σ ∗ (β1∗ (β2∗ ),β2∗ )
≥ J2 β1∗ (β2 ), β2 , σ ∗ (β1∗ (β2 ), β2 ) , (15)

where ties in (13) are broken as in Section II-A. Under this tie-breaking rule, (13) yields the response (12). Since J1 + J2 = 1, all maximizers in (14) give the leader the same payoff, so no tie-breaking rule is needed for the follower, and (15) states that β2∗ minimizes the follower’s optimal payoff. With a slight abuse of notation, we also write (β1∗ , β2∗ , σ ∗ ) for the equilibrium outcome, with β1∗ = β1∗ (β2∗ ) and σ ∗ = σ ∗ (β1∗ , β2∗ ). III. E QUILIBRIUM C HARACTERIZATION

We characterize the equilibria by backward induction. Taking the receiver’s response (12) as given, we first derive sender 1’s best response (14) to a fixed leader policy and then determine sender 2’s optimal commitment (15).

B. Sender 2’s Optimal Policy

A. Sender 1’s Best Response +

Fix β2 = (s2 , c2 ) ∈ B2 and let δ := (cmin − c2 ) , where (z)+ := max{z, 0}. By (11), whenever s1 + s2 > 0, the PC holds if and only if c1 ≥ δ. Thus, δ is the minimum state-0 intensity that sender 1 must supply for the receiver to follow the reports. Lemma 1: For every β2 = (s2 , c2 ) ∈ B2 , the policy ( (0, 0), R1 < δ, ∗ β1 (β2 ) = (16) (R1 − δ, δ), R1 ≥ δ, with δ = (cmin −c2 )+ , satisfies (14). If the follower’s optimal payoff is positive, β1∗ (β2 ) is the unique maximizer. If the optimal payoff is zero, every β1 ∈ B1 is a maximizer; we retain the selection in (16) in the subsequent calculations. Proof: For a given feasible leader policy β2 = (s2 , c2 ), (11) implies that, whenever s1 + s2 > 0, the PC holds if and only if c1 ≥ δ. Thus, δ is the minimum state-0 reporting intensity that sender 1 must supply for the receiver to follow the reports. Suppose R1 < δ. Then every β1 ∈ B1 satisfies c1 ≤ R1 < δ. If s1 + s2 > 0, the PC fails, so the receiver selects σd and sender 1 obtains a zero payoff. If s1 + s2 = 0, the estimate remains 0, and sender 1 again obtains a zero payoff. Hence, every β1 ∈ B1 , including (0, 0), satisfies (14). Suppose R1 ≥ δ. Consider any β1 ∈ B1 with s1 + s2 > 0 that satisfies the PC. By the PC and the budget, c1 ≥ δ and s1 ≤ R1 − c1 ≤ R1 − δ. Since s1 + s2 > 0 and δ + c2 = max{cmin, c2 } > 0, [3, Lemma 1] gives J1 (s1 + s2 , c1 + c2 ) ≤ J1 (s1 + s2 , δ + c2 ) ≤ J1 (R1 − δ + s2 , δ + c2 ),

(17)

where the first step lowers the state-0 intensity to its minimum admissible value and the second raises the state-1 intensity to its maximum feasible value. The policy (R1 − δ, δ) is feasible because R1 ≥ δ, satisfies the PC since δ + c2 ≥ cmin , and attains the right-hand side of (17). Since policies that violate the PC or have s1 + s2 = 0 yield a zero payoff, (R1 − δ, δ) satisfies (14). If the follower’s optimal payoff is positive, every maximizer has s1 + s2 > 0 and satisfies the PC, so it obeys (17). Because the monotonicity in [3, Lemma 1] is strict, both inequalities in (17) must then hold with equality, which gives c1 = δ and s1 = R1 − δ and hence the maximizer is unique. If the optimal payoff is zero, every β1 ∈ B1 is a maximizer, since all payoffs are nonnegative. Lemma 1 shows how the leader’s policy shapes the follower’s allocation. Since δ = (cmin − c2 )+ decreases in c2 until it vanishes at c2 = cmin , state-0 reports by sender 2 reduce the intensity that sender 1 must devote to participation and leave more of its budget for state-1 reports. At the boundary R1 = δ, the selection in (16) is (0, δ), so the follower’s own state-1 intensity is zero. If s2 > 0, this selection is the unique best response and yields a positive payoff, because the aggregate state-0 intensity δ + c2 ≥ cmin satisfies the PC and makes the leader’s state-1 reports acceptable to the receiver.

By (15) and J1 +J2 = 1, sender 2 minimizes the follower’s optimal payoff. For R1 > cmin and c2 ≥ 0, let V (c2 ) denote the follower’s payoff when the leader uses β2 = (0, c2 ). The next lemma shows that the leader need only compare silence with full-budget state-0 reporting. Lemma 2: There exists β2∗ = (s∗2 , c∗2 ) satisfying (15) with ∗ s2 = 0. If R1 ≤ cmin , the silent policy (0, 0) satisfies (15). If R1 > cmin , every β2∗ satisfying (15) has s∗2 = 0 and c∗2 ∈ {0, R2 }. Proof: Fix β2 = (s2 , c2 ) ∈ B2 and let δ = (cmin −c2 )+ . If R1 < δ, the follower’s optimal payoff is zero for every s2 by Lemma 1. Otherwise, the best response (16) induces the aggregate rates s = R1 + s2 − δ,

c = max{cmin , c2 },

(18)

and, since c > 0, the follower’s optimal payoff J1 (s, c) is strictly increasing in s2 by [3, Lemma 1]. In both cases, replacing (s2 , c2 ) with (0, c2 ) preserves feasibility and does not increase the follower’s optimal payoff, so it suffices to consider s2 = 0. If R1 > cmin , then R1 > δ for every c2 , and this replacement strictly decreases the follower’s optimal payoff whenever s2 > 0, and thus every β2∗ satisfying (15) has s∗2 = 0. If R1 ≤ cmin , the silent policy leaves the follower with a zero optimal payoff by Lemma 1, so it attains J2 = 1, which is the largest possible leader payoff. Now suppose R1 > cmin and s2 = 0. By (18), we have ( J1 (R1 −cmin +c2 , cmin ), 0 ≤ c2 ≤ cmin , V (c2 ) = (19) J1 (R1 , c2 ), c2 ≥ cmin . As c2 increases, only the state-1 intensity grows in the first branch and only the state-0 intensity grows in the second. Hence, by [3, Lemma 1], we obtain V ′ (c2 ) > 0, 0 < c2 < cmin , V ′ (c2 ) < 0, c2 > cmin .

(20)

Since the two branches agree at c2 = cmin , V (c2 ) is continuous, strictly increasing on [0, cmin], and strictly decreasing on [cmin , ∞). Its minimum over the feasible set [0, R2 ] therefore exists and is attained only at c2 ∈ {0, R2 }. Together with the reduction to s2 = 0, this proves all three claims. The proof of Lemma 2 identifies two opposing effects of the leader’s state-0 reports. For c2 < cmin , additional reports free part of the follower’s budget for state-1 reports while keeping the aggregate state-0 intensity at cmin , which benefits the follower. For c2 ≥ cmin , the follower already devotes its entire budget to state-1 reports, so additional state0 reports benefit the leader. Hence, the leader strictly prefers full-budget reporting to silence if and only if V (R2 ) < V (0). Two budget thresholds, defined in Theorem 1, determine the equilibrium. The first, R1crit , is the follower budget above which intervention can be strictly preferred: if R1 ≤ R1crit , silence is optimal for every R2 . For R1 > R1crit , the second, R2crit , is the leader budget above which intervention is strictly preferred, with indifference at R2 = R2crit .

Theorem 1: Let φ(R1 ) := R1 (R1 + λ + µ − cmin ) and ψ := cmin (cmin + λ + µ). Let R1crit be the unique solution of φ(R1 ) = ψ in (cmin , ∞) given by 1 cmin − λ − µ R1crit = 2 q  (21) + (λ + µ)2 + 2cmin(λ + µ) + 5c2min ,

and, for R1 > cmin , let   cmin φ(R1 ) , R1 > Rcrit , 1 crit R2 := φ(R1 ) − ψ  +∞, R1 ≤ R1crit ,

(22)

Then Os is the unique equilibrium outcome if R2 < R2crit , Oi is the unique equilibrium outcome if R2 > R2crit , and these two are the only equilibrium outcomes if R2 = R2crit . Proof: Part 1. Suppose R1 ≤ cmin , and fix β2 = (s2 , c2 ) ∈ B2 with δ = (cmin − c2 )+ . By Lemma 1, the follower’s optimal payoff is zero if R1 < δ. If R1 ≥ δ, it equals J1 (R1 −δ+s2, max{cmin , c2 }), which by (8) vanishes if and only if R1 − δ + s2 = 0, that is, R1 = δ and s2 = 0. Silence satisfies this condition, so the minimum follower payoff is zero, and β2 satisfies (15) if and only if the stated condition holds. For each such β2 , every β1 ∈ B1 is a maximizer by Lemma 1 and yields J1 = 0 and J2 = 1. With β1 = β2 = (0, 0), the receiver selects σd by (12). Part 2. Suppose R1 > cmin . By Lemma 2, the leader’s candidates are (0, 0) and (0, R2 ). If R2 ≤ cmin , then V (R2 ) > V (0) unless R2 = 0, because V is strictly increasing on [0, cmin ], silence is the unique optimal policy, consistent with R2 < R2crit . Now let R2 > cmin . By (8) and (19), we have

V (R2 ) = J1 (R1 , R2 ) =

J2

R2crit c2

cmin (a)

so that R2crit > cmin . The pure-strategy equilibrium outcomes are as follows. 1) If R1 ≤ cmin , the equilibrium outcomes are the triples (β1 , β2 , σ ∗ (β1 , β2 )) with β1 ∈ B1 and β2 = (s2 , c2 ) ∈ B2 such that either R1 < δ, or R1 = δ and s2 = 0, where δ := (cmin − c2 )+ . All of them yield J1 = 0 and J2 = 1; in particular, ((0, 0), (0, 0), σd ) is an equilibrium outcome. 2) If R1 > cmin , we define the silence and intervention outcomes as
Os := (R1 − cmin , cmin ), (0, 0), σs ,
Oi := (R1 , 0), (0, R2 ), σs .

V (0) = J1 (R1 − cmin , cmin ) =

J2

λN0 , (λ + µ)D0

λN2 , (λ + µ)D2

where N0 := (R1 − cmin )(cmin + λ + µ), D0 := µcmin + (λ + cmin )(R1 − cmin ), N2 := R1 (R2 + λ + µ), and D2 := µR2 + (λ + R2 )R1 are positive. Direct expansion gives   N0 D2 − N2 D0 = µ (φ(R1 ) − ψ)R2 − cmin φ(R1 ) , (23)

c2

cmin (b)

Fig. 2: Schematic illustration of sender 2’s payoff 1 − V (c2 ) under β2 = (0, c2 ), given the follower’s best response, for (a) cmin < R1 < R1crit and (b) R1 > R1crit . The payoff decreases on [0, cmin ] and increases thereafter. The dashed line marks the silence payoff 1 − V (0). In (a), every c2 > 0 yields less than silence. In (b), the payoff exceeds that of silence exactly when c2 > R2crit , so full-budget reporting is optimal if R2 > R2crit . so V (R2 ) is smaller than, equal to, or larger than V (0) according as the bracket is positive, zero, or negative. The function φ − ψ equals −c2min at cmin and has derivative 2R1 + λ + µ − cmin > 0 for R1 ≥ cmin . Hence, it has the unique root R1crit in (cmin , ∞), given by (21), and is positive exactly when R1 > R1crit . If R1 ≤ R1crit , the bracket is negative for every R2 , so silence is strictly optimal. If R1 > R1crit , the bracket has the sign of R2 − R2crit , where R2crit > cmin because ψ > 0. Thus, silence is the unique optimal policy if R2 < R2crit , full-budget state-0 reporting is the unique optimal policy if R2 > R2crit , and both are optimal if R2 = R2crit . Finally, under silence, δ = cmin < R1 and the follower’s optimal payoff is positive, so its unique best response is (R1 −cmin , cmin ). Under optimal intervention, R2 ≥ R2crit > cmin , so δ = 0 and its unique best response is (R1 , 0). In both cases, s > 0 and c ≥ cmin , so the receiver selects σs by (12). This yields Os and Oi . Fig. 2 illustrates the leader’s choice between silence and intervention. At R2 = R2crit , Os and Oi yield identical sender payoffs but, as Section IV shows, different receiver utilities. IV. R ECEIVER B ENEFIT FROM C OMPETITION As a benchmark, we consider a single sender with sender 1’s preferences and total budget R := R1 + R2 , with the source, receiver, and budget model unchanged. This benchmark coincides with our game for (R1 , R2 ) = (R, 0), in which the leader can only remain silent. By [3, Theorem 1], or by applying Theorem 1 with R2 = 0, the receiver’s utility under this benchmark is qµ single JR (R) = qπ0 = , R ≥ 0. (24) λ+µ If the single sender instead had sender 2’s preferences, it could not gain from reporting, because the receiver’s default estimate 0 is already the one it prefers. An optimal policy for such a sender is therefore to remain silent, in which case the estimate remains 0 and the receiver again obtains qπ0 .

(25)

with equality at every equilibrium outcome other than Oi . The outcome Oi is an equilibrium outcome if and only if R1 > cmin and R2 ≥ R2crit , in which case ∗ JR − qπ0 =

(1 − q)λR1 (R2 − cmin ) > 0. (λ + µ)(µR2 + λR1 + R1 R2 )

0.34

Equilibrium Single-sender baseline Large-budget limit

0.32

0

5

10

15

20

R2 ∗ Fig. 3: Equilibrium receiver utility JR versus R2 for R1 = 2, crit where R2 = 2. The two marked points at R2 = 2 are the utilities under Os and Oi . The horizontal lines show the single-sender benchmark qπ0 = 0.3 and the limit 0.38 as R2 grows.

0.8 0.7

(26)

In particular, at R2 = R2crit , the two equilibrium outcomes give the senders identical payoffs but give the receiver qπ0 under Os and qπ0 plus the strictly positive gain in (26) under Oi . Proof: By (13), the receiver’s equilibrium utility is at least qπ0 , the utility it obtains under σd . Together with (24), this proves (25). By Theorem 1, every equilibrium outcome other than Oi either has R1 ≤ cmin or equals Os . In the first case, J1 = 0 in equilibrium, whereas (8) gives J1 (s, c) > 0 whenever the receiver follows reports with s > 0. Thus, the receiver either selects σd or follows reports with s = 0, and in both ∗ cases JR = qπ0 . Under Os , the aggregate rates are (s, c) = ∗ (R1 − cmin , cmin ), so (10) gives JR = qπ0 . By Theorem 1, Oi is an equilibrium outcome if and only if R1 > cmin and R2 ≥ R2crit . Its aggregate rates are (s, c) = (R1 , R2 ), so (10) gives (26), which is positive because R2 ≥ R2crit > cmin . Finally, at R2 = R2crit , both outcomes are equilibria. They give the senders identical payoffs, since the leader is indifferent between them, but give the receiver qπ0 under Os and qπ0 plus the strictly positive gain in (26) under Oi . The gain arises because the opposing sender makes the PC slack. A single sender supplies only the state-0 intensity cmin that participation requires, so the PC binds. Under Oi , sender 2 supplies c2 = R2 > cmin , while sender 1 already sets c1 = 0 and therefore cannot offset these reports by reducing its own state-0 intensity. Hence, competition raises the receiver’s utility even though the total budget is unchanged. For fixed R1 > R1crit , the equilibrium receiver-utility correspondence has distinct branches on either side of R2crit . For R2 < R2crit , the unique equilibrium yields qπ0 , whereas for R2 > R2crit , the unique intervention equilibrium yields the strictly larger utility in (26). At R2 = R2crit , both values are equilibrium utilities. The sender payoffs, in contrast, are

R2crit 0.36

0.3

utility

single ∗ (R1 + R2 ) = qπ0 , JR ≥ JR

0.38

receiver utility

For the benchmark sender with sender 1’s preferences, if R ≤ cmin , the sender cannot obtain a positive payoff while satisfying the PC, whereas for R > cmin it chooses (s, c) = (R−cmin , cmin ), which makes the PC bind. In both cases, the receiver obtains only its default utility, even though a single sender controls the entire budget. More generally, by (10), ZOH strictly improves on the default utility exactly when s > 0 and c > cmin , and the gain is zero when s = c = 0. Competition can therefore benefit the receiver only through an equilibrium with s > 0 and c > cmin . ∗ Theorem 2: Let JR denote the receiver’s utility at an equilibrium outcome of Theorem 1. For all R1 , R2 ≥ 0,

0.6

J2∗

0.5

J1∗

0.4

Silent-leader baseline (J2 ) Large-budget limits

0.3 0.2 0

5

10

15

20

R2

Fig. 4: Equilibrium sender payoffs J1∗ and J2∗ versus R2 for R1 = 2, where R2crit = 2. continuous across the threshold because Os and Oi give them identical payoffs there. V. N UMERICAL R ESULTS We consider a binary source with λ = 2, µ = 3, and q = 0.5, which satisfy qπ0 > (1 − q)π1 . These parameters give cmin√ = 1, default receiver utility qπ0 = 0.3, and R1crit = 10 − 2 ≈ 1.1623. All equilibrium curves follow from Theorem 1, and the receiver’s utilities from Theorem 2. We first fix R1 = 2 and vary R2 , so that φ(2) = 12, ψ = 6, and R2crit = 2. As Fig. 3 shows, the receiver obtains qπ0 = 0.3 for R2 < 2, where Os is the unique equilibrium outcome. At R2 = 2, both Os and Oi are equilibrium outcomes, giving the receiver 0.3 and 0.3286, respectively. For R2 > 2, the receiver’s utility under Oi increases with R2 and approaches 0.38 as R2 grows, illustrating the strict gain in Theorem 2. Fig. 4 shows the corresponding sender payoffs. For R2 ≤ 2, they equal J1∗ = V (0) = 0.4 and J2∗ = 0.6. For R2 > 2, the follower’s payoff V (R2 ) decreases and approaches 0.16 as R2 grows, while the leader’s payoff increases toward 0.84. The sender payoffs are thus continuous at R2 = 2, whereas the receiver’s utility is not. Finally, Fig. 5 fixes R1 + R2 = 8 and varies the budget division. We compare the equilibrium with a receiver-optimal benchmark that chooses the aggregate intensities directly, under the same budget and the same choice between ZOH

receiver utility

0.37 0.36 0.35 0.34

Competing senders Single sender Coordinated benchmark Best equilibrium split

0.33 0.32 0.31 0.3 0

2

R1

4

6

8

∗ Fig. 5: Equilibrium receiver utility JR versus the budget division (R1 , 8 − R1 ), compared with the receiver-optimal opt benchmark JR (8) ≈ 0.3676.

and the default estimator: opt JR (8) = max max{qπ0 , JR (s, c)}. s,c≥0 s+c≤8

(27)

By (10), JR (s, c) is strictly increasing in c whenever s > 0, since its c-derivative has the sign of λs + cmin (µ + s) > 0. Hence, any maximizer above the default utility satisfies s + c = 8. Substituting c = 8 − s gives JR (s, 8 − s) = 0.3 +

s(7 − s) , 5[24 + s(7 − s)]

which is increasing in s(7 − s) and hence maximized at s = 3.5. The benchmark is therefore attained at (s, c) = opt (3.5, 4.5), with JR (8) ≈ 0.3676. At (R1 , R2 ) = (3.5, 4.5), we have R2 > R2crit ≈ 1.30, so Oi is the unique equilibrium outcome: sender 1 reports only state 1, sender 2 reports only state 0, and the aggregate intensities (s, c) = (3.5, 4.5) coincide with the benchmark maximizer. More generally, since Oi implements (s, c) = (R1 , R2 ), any benchmark maximizer (s, c) satisfying s > R1crit and c > R2crit (s) is uniquely implemented by the corresponding budget division (R1 , R2 ) = (s, c), where R2crit (s) denotes (22) evaluated at R1 = s. At c = R2crit (s), the same reporting rates are attained under the intervention equilibrium, but the equilibrium outcome is not unique. Moving away from (3.5, 4.5) reduces the receiver’s utility, and if either sender receives too little budget, the receiver obtains only its default utility 0.3, as under a single sender with the entire budget. Thus, in this example, an appropriate budget division allows strategic reporting to attain receiver-optimal performance within the specified estimator architecture. VI. C ONCLUSION We studied competition between two senders that influence a receiver through the timing of truthful reports about a binary continuous-time Markov source. We characterized the Stackelberg equilibria in closed form. When the follower can obtain positive utility against a silent leader, the leader’s state-0 reports initially help the follower by reducing the reporting effort needed for participation, so every optimal leader policy is either silence or full-budget state-0 reporting, with explicit budget thresholds determining which. In

the low-budget regime, equilibrium leader policies can be nonunique, but all such equilibria leave the receiver at its default utility. When the full-budget intervention outcome is optimal, it makes the participation constraint slack and strictly improves the receiver’s utility relative to a single sender with the same total budget. In our numerical example, an appropriate budget division attains a receiver-optimal reporting benchmark. Future work will consider opposing senders that allocate their budgets across multiple heterogeneous sources, alternative commitment orders, and receivers that also exploit information conveyed by the absence of reports. R EFERENCES [1] S. Kaul, R. Yates, and M. Gruteser, “Real-time status: How often should one update?,” in 2012 Proceedings IEEE INFOCOM, pp. 2731– 2735, 2012. [2] R. D. Yates, Y. Sun, D. R. Brown, S. K. Kaul, E. Modiano, and S. Ulukus, “Age of information: An introduction and survey,” IEEE Journal on Selected Areas in Communications, vol. 39, no. 5, pp. 1183–1210, 2021. [3] A. B. Gundogan and M. Bastopcu, “Strategic persuasion through information timeliness,” arXiv preprint arXiv:2607.15939v1, 2026. [4] Y. Sun, Y. Polyanskiy, and E. Uysal, “Sampling of the wiener process for remote estimation over a channel with random delay,” IEEE Transactions on Information Theory, vol. 66, no. 2, pp. 1118–1135, 2020. [5] M. Bastopcu and S. Ulukus, “Using timeliness in tracking infections,” Entropy, vol. 24, no. 6, p. 779, 2022. [6] N. Akar and S. Ulukus, “Timely monitoring of Markov chains under sampling rate constraints,” in ICC 2024 - IEEE International Conference on Communications, pp. 189–194, 2024. [7] E. Kamenica and M. Gentzkow, “Bayesian persuasion,” American Economic Review, vol. 101, no. 6, pp. 2590–2615, 2011. [8] D. Bergemann and S. Morris, “Information design: A unified perspective,” Journal of Economic Literature, vol. 57, no. 1, pp. 44–95, 2019. [9] P. H. Au, “Dynamic information disclosure,” RAND Journal of Economics, vol. 46, pp. 791–823, October 2015. [10] J. Renault, E. Solan, and N. Vieille, “Optimal dynamic information provision,” Games and Economic Behavior, vol. 104, pp. 329–349, 2017. [11] G. Ashkenazi-Golan, P. Hernández, Z. Neeman, and E. Solan, “Markovian persuasion with two states,” Games and Economic Behavior, vol. 142, pp. 292–314, 2023. [12] E. Lehrer and D. Shaiderman, “Markovian persuasion with stochastic revelations,” Games and Economic Behavior, vol. 154, pp. 411–439, 2025. [13] J. C. Ely, “Beeps,” American Economic Review, vol. 107, no. 1, pp. 31– 53, 2017. [14] F. Farhadi and D. Teneketzis, “Dynamic information design: A simple problem on optimal sequential information disclosure,” Dynamic Games and Applications, vol. 12, no. 2, pp. 443–484, 2022. [15] R. Aı̈d, O. Bonesini, G. Callegaro, and L. Campi, “Continuous-time persuasion by filtering,” Journal of Economic Dynamics and Control, vol. 176, p. 105100, 2025. [16] Y.-K. Che, K. Kim, and K. Mierendorff, “Keeping the listener engaged: A dynamic model of Bayesian persuasion,” Journal of Polit. Econ., vol. 131, no. 7, pp.1797–1844, 2023. [17] M. Gentzkow and E. Kamenica, “Competition in persuasion,” The Review of Economic Studies, vol. 84, no. 1, pp. 300–322, 2017. [18] M. Gentzkow and E. Kamenica, “Bayesian persuasion with multiple senders and rich signal spaces,” Games and Economic Behavior, vol. 104, pp. 411–429, 2017. [19] P. H. Au and K. Kawai, “Competitive information disclosure by multiple senders,” Games and Economic Behavior, vol. 119, pp. 56– 78, 2020. [20] F. Li and P. Norman, “Sequential persuasion,” Theoretical Economics, vol. 16, no. 2, pp. 639–675, 2021. [21] R. K. Velicheti, M. Bastopcu, and T. Başar, “Value of information in games with multiple strategic information providers,” IEEE Transactions on Automatic Control, vol. 70, no. 7, pp. 4532–4547, 2025.

Related documents

• Timely Persuasion with Opposing Senders: When Does Competition Help the Receiver?#1200119
• Opening up new possibilities for algorithmic fairness#85595
• Defense Feedback and Global-Game Selection#1154946
• Preference-based Cooperation in a Prisoner's Dilemma Game: Whole Population Cooperation without Information Flow across Matches#1276045
• Monitoring algorithmic fairness in sequential decision making#1146782